Numerical Methods for Singularly Perturbed Boundary Value Problems Modeling Diffusion Processes

Kolmogorov, V. L.; Шишкин, Г. И.

doi:10.1002/9780470141564.ch3

Cited by 8 publications

(8 citation statements)

References 14 publications

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“…It is known that even in the case of singularly perturbed problems in bounded domains with sufficiently smooth solutions to the problem, solutions obtained by classic difference schemes do not converge ε-uniformly (see, e.g., [3,5] Our aim is to construct a difference scheme for problem (1.2), (1.1) convergent ε-uniformly in the weighted norm · w on grids introduced in G. In addition, it is required to construct a constructive difference scheme on grids with a bounded number of nodes allowing us to approximate the solution to the problem in the weighted norm ε-uniformly in bounded domains given on G.…”

Section: 2mentioning

confidence: 99%

Constructive and formal difference schemes for singularly perturbed parabolic equations in unbounded domains in the case of solutions growing at infinity

Шишкин

2009

Russian Journal of Numerical Analysis and Mathematical Modelling

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An initial boundary value problem for a singular perturbed parabolic reaction-diffusion equation is considered in a domain unbounded in x on the real axis; the leading derivative of the equation contains the parameter ε 2 ; ε ∈ (0, 1]. The right-hand side of the equation and the initial function indefinitely grow as O x 2 for x → ∞, which leads to an indefinite growth of the solution at infinity as O Ψ(x) , where Ψ(x) = x 2 + 1. For small values of the parameter ε a parabolic boundary layer appears in the neighbourhood of the lateral part of the boundary. In this problem for fixed values of the parameter ε the error of the grid solution indefinitely grows in the uniform norm for x → ∞. The closeness of the solutions to the initial boundary value problem and to its grid approximations is considered in this paper in the weighted uniform norm · w with the weight function Ψ −1 (x); the solution to the initial boundary value problem is ε-uniformly bounded in this norm. Using special grids condensing in the neighbourhood of the boundary layer, we construct formal difference schemes (schemes on grids with an infinite number of nodes) convergent ε-uniformly in the weighted norm. Based on the notion of the domain of essential dependence of the solution, we propose constructive difference schemes (schemes on grids with a finite number of nodes) convergent ε-uniformly in the weighted norm on given bounded subdomains.Special numerical methods for solving boundary value problems for singularly perturbed equations in bounded domains are well elaborated for the case of sufficiently smooth data (see, e.g., [2 -4, 8, 9, 13, 20] and references therein). Such numerical methods allow one to approximate solutions with an error not depending on the perturbation parameter ε, as the methods converge ε-uniformly.Only a few publications have considered ε-uniformly convergent methods for problems in unbounded domains. Special difference schemes for singularly perturbed problems for elliptic and parabolic equations in unbounded domains were studied in [14,18,19] and [1], respectively; the solutions to those problems were ε-uniformly bounded.

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Section: 2mentioning

confidence: 99%

Constructive and formal difference schemes for singularly perturbed parabolic equations in unbounded domains in the case of solutions growing at infinity

Шишкин

2009

Russian Journal of Numerical Analysis and Mathematical Modelling

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“…The smallness of these parameters induces boundary, initial and initial-boundary layers in the solution of the problem. As a consequence, for such problems the errors in the discrete solutions that are obtained using classical difference schemes are commensurable with the solutions of the boundary value problem itself (see, e.g., [16]) just as for parabolic problems where the highest-order spatial derivatives are multiplied by a small parameter [2,3,7,9,10,15].…”

Section: Introductionmentioning

confidence: 99%

Finite Difference Scheme for a Singularly Perturbed Parabolic Equations in the Presence of Initial and Boundary Layers

Cordero¹,

Cronin

Шишкин

et al. 2008

Mathematical Modelling and Analysis

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The grid approximation of an initial-boundary value problem is considered for a singularly perturbed parabolic reaction-diffusion equation. The secondorder spatial derivative and the temporal derivative in the differential equation are multiplied by parameters ε 2 1 and ε 2 2 , respectively, that take arbitrary values in the open-closed interval (0, 1]. The solutions of such parabolic problems typically have boundary, initial layers and/or initial-boundary layers. A priori estimates are constructed for the regular and singular components of the solution. Using such estimates and the condensing mesh technique for a tensor-product grid, piecewise-uniform in x and t, a difference scheme is constructed that converges ε-uniformly at the rate O(N −2 ln 2 N + N −1 0 ln N0), where (N + 1) and (N0 + 1) are the numbers of mesh points in x and t respectively.

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“…Problems for singularly perturbed parabolic reaction-diffusion equations with a strong singularity (a discontinuous initial condition) were considered in [6,7,8,9,10,11,12]. In these problems, when constructing special schemes, in addition to the condensing mesh method, a specific technique was used, such as the fitted operator method in [7,8,9,10] or the method of additive splitting of singularities in [6,11,12] (in a neighbourhood of the points of discontinuity to the initial function).…”

Section: Introductionmentioning

confidence: 99%

“…In these problems, when constructing special schemes, in addition to the condensing mesh method, a specific technique was used, such as the fitted operator method in [7,8,9,10] or the method of additive splitting of singularities in [6,11,12] (in a neighbourhood of the points of discontinuity to the initial function). In the fitted operator method, the singular components of the solution (or some of them) are solutions of a difference scheme, and in the method of additive splitting of singularities, these components are included in the approximate solution additively (for a description of this method in the case of a regular problem with nonsmooth data, see, e.g., [13] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

Improved Difference Scheme for a Singularly Perturbed Parabolic Reaction-Diffusion Equation with Discontinuous Initial Condition

Shishkin

2009

Lecture Notes in Computer Science

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Abstract.A Dirichlet problem is considered for a singularly perturbed parabolic reaction-diffusion equation with a piecewise-continuous initial condition. For this problem, using the method of additive splitting of singularities (generated by discontinuities of the initial function and its lowest derivatives) and the Richardson method, a finite difference scheme on piecewise-uniform meshes is constructed that converges ε-uniformly with the third and second order of accuracy in x and t, respectively.

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Numerical Methods for Singularly Perturbed Boundary Value Problems Modeling Diffusion Processes

Cited by 8 publications

References 14 publications

Constructive and formal difference schemes for singularly perturbed parabolic equations in unbounded domains in the case of solutions growing at infinity

Constructive and formal difference schemes for singularly perturbed parabolic equations in unbounded domains in the case of solutions growing at infinity

Finite Difference Scheme for a Singularly Perturbed Parabolic Equations in the Presence of Initial and Boundary Layers

Improved Difference Scheme for a Singularly Perturbed Parabolic Reaction-Diffusion Equation with Discontinuous Initial Condition

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